09-09-2026, 09:54 PM
Spherical trigonometry studies the relationships between the sides and angles of triangles drawn on the surface of a sphere. Unlike ordinary plane triangles, the sides of a spherical triangle are arcs of great circles, which are the geodesics of a sphere. This geometry is important in astronomy, geodesy, navigation, and calculations involving positions on Earth. A fundamental difference from Euclidean geometry is that the angles $A,B,C$ of a spherical triangle satisfy $A+B+C>\pi$. For a sphere of radius $R$, side lengths are usually expressed as angular quantities by dividing their arc lengths by $R$.
The central formulas are spherical versions of the familiar sine and cosine laws. The spherical law of cosines is $\cos a=\cos b\cos c+\sin b\sin c\cos A$, with analogous formulas obtained by cyclically permuting $a,b,c$. The spherical law of sines is $\frac{\sin A}{\sin a}=\frac{\sin B}{\sin b}=\frac{\sin C}{\sin c}$. For triangles that are very small compared with the sphere's radius, these formulas approach the ordinary Euclidean sine and cosine laws. Spherical trigonometry also includes Napier's analogies, Delambre's analogies, half-angle formulas, and special rules for right spherical triangles.
One of the most striking consequences of spherical geometry is the connection between area and angle sum. If a spherical triangle has angles $A,B,C$, its spherical excess is $E=A+B+C-\pi$. On a unit sphere, $E$ is exactly the area of the triangle. More generally, on a sphere of radius $R$, the area is $\text{Area}=R^2(A+B+C-\pi)=R^2E$. Thus, unlike in Euclidean geometry, the angle sum of a spherical triangle directly determines its area.
Key takeaways:
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The central formulas are spherical versions of the familiar sine and cosine laws. The spherical law of cosines is $\cos a=\cos b\cos c+\sin b\sin c\cos A$, with analogous formulas obtained by cyclically permuting $a,b,c$. The spherical law of sines is $\frac{\sin A}{\sin a}=\frac{\sin B}{\sin b}=\frac{\sin C}{\sin c}$. For triangles that are very small compared with the sphere's radius, these formulas approach the ordinary Euclidean sine and cosine laws. Spherical trigonometry also includes Napier's analogies, Delambre's analogies, half-angle formulas, and special rules for right spherical triangles.
One of the most striking consequences of spherical geometry is the connection between area and angle sum. If a spherical triangle has angles $A,B,C$, its spherical excess is $E=A+B+C-\pi$. On a unit sphere, $E$ is exactly the area of the triangle. More generally, on a sphere of radius $R$, the area is $\text{Area}=R^2(A+B+C-\pi)=R^2E$. Thus, unlike in Euclidean geometry, the angle sum of a spherical triangle directly determines its area.
Key takeaways:
- Spherical triangles are formed by arcs of great circles.
- Their angles satisfy $A+B+C>\pi$, unlike Euclidean triangles where the sum is $\pi$.
- The spherical sine law and spherical cosine law are the main tools for solving spherical triangles.
- The spherical excess $E=A+B+C-\pi$ measures how much the angle sum exceeds $\pi$.
- The area of a spherical triangle is $\text{Area}=R^2E$.
- Spherical trigonometry has important applications in astronomy, navigation, geodesy, and calculations on the Earth's surface.
ARTICLE / ARTICLE [PDF]
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